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CONSTRUCTION OF COST-OF-LIVING INDEX NUMBERS
- A UNIFIED APPROACH
D.S. Prasada Rao
No.5 - April 1979
ISSN 0157-0188
ISBN 0 85834 235 9
CONSTRUCTION OF COST-OF-LIVING INDEX NUMBERS - AUNIFIED APPROACH
D.S. PRASADA RAO
University of New England, Armidale, Australia.
There are a number of alternative index number’formulae available
in literatttre that can be used to measure changes in general price level.
Generally these index numbers can be used only for binary comparisons,
i.e., comparison of two price vectors prevailing in two situations. These
indices then can be chained to obtain price comparisons when more than two
price vectors are involved in comparisons.
Since there is a multitude of these index number formulae, generally
choice of an ’ideal’ index number formula turns out to be a formidable
problem. All the usual tests to separate the good formulae from the rest
are generally not of much use either because there is generally a large
subset of these formulae which still satisfy these tests or because these
tests themselves turn out to be arbitrary and one may find it hard.to reject
an index number formula just because it does not satisfy these tests.
An alternative approach to this problem is to look at.all these index
number formulae as special cases of a much general structure. In such a
case each index number.formula can be judged according ~to the special
restrictions necessary to derive the formula from the general structure.
Then the choice problem can be viewed as one Of evaluating these special
conditions. This seems to be a logical way of tackling the problem of choice.
In what follows, in section i, we discuss two such general approaches
to construction of index numbers which are already in vogue. In section 2,
we present a new unified approach to the index number construction. Then
we derive some of the well-known index number formulae such as Laspeyres and
Paasche index numbers. In this section we also consider some possible ways
of generalizing these index number formulae for multilateral comparisons.
In the last section we discuss briefly a more general system involving
basic micro-economic theoretic ideas.
SECTION I. Two known approaches
We shall discuss these two approaches and rest of this paper with
fol!owing notation. Let (pl,ql), (p2,q2)...(pM,qM) represent M pairs of
price and quantity vectors of dimension N, the number of commodities. Then
Pij and qij stand for price and quantity, respectively, of ith commodity in
jth pair of price and quantity vectors.~ These M pairs may represent price
and quantity observations over time or space. For purposes of exposition
we consider these as observations over time. The problem is one of binary
comparisons if M = 2 and M ~ 3 concerns multilateral comparisons. As noted
earlier, one can always construct indices for multilateral comparisons
using the chain method and index numbers for~binary comparisons.
Let us briefly look at the two general approaches which are discussed
usually in literature. We shall cal! them statistical and welfare approaches
to construction of cost-of-living index numbers.
i.i Statistical approach: This approach makes us’e of the idea of measuring
price change for one commodity. For any commodity, say ith, price change
can be measured by the ratio of prices in two periods, say 1 and 2, which
is given by Pi2/Pil. This ratio measures the price change f-~om period 1 to
period 2, for ith commodity. This is known as a price relative. Thus we
have N such price relatives Pi2/Pil, for i = i, 2,..., N. Now the price
index number construction can be posed as a problem of combining these N
ratios into a single numerical value. This can be viewed as a statistica!
problem of obtaining a suitable ’measure of location (Central tendency)’
Now most of the known index number formulae can be shown to be different
measures of central tendency. One can think of arithmetic, geometric or
-3harmonic mean of these price relatives to be appropriate means of calculating
.locatlon parameter. Then there is choice of weighted or unweighted means
an~ one can think of some suitable functions of these basic index numbers
tOO.
Laspeyres, Paasche, Edgeworth-Marshall (see Fisher (1922) Geary-Khamis
(Geary (1958), Khamis (1970), Braithwaite (1968)) fixed weight index numbers
can be looked at as special cases of weighted arithmetic means of the price
N
[~Pi___~ .w. where w.’s
are such that 0 ~ w.~ S i
~
i:l Pil
~
relatives, which is given by
N
and ~ w. : i. Then
i=l 1
Pilqil
~Pi2qil
can be derived with w. =
Laspeyres index =
l ZPilqil
ZPilqil
ZPi2qi2
Paasche index :
ZPilqi2
Pilqi2
can be derived with w. l Z~qi2
ZPi2(qil+qi2)
Edgeworth-Marshall index -
Pil(qil+qi2)
with
w.’
’
x ZPil(qil+qi2)
ZPil(qil+qi2)
qilqi2
ZPi2 qil+qi2
Geary-Khamis index =
ZPil
qilqi2
qil+qi2
ZPi2qia
Braithwaite fixed index :
qilqi2
Pll qil+qi2
with w.
1
with w. -
ZPilqia
qilqi2
ZPil qil+qi2
Pilqia
ZPilqia
where qa is an arbitrarily chosen quantity vector.
Similarly Kloek-Theil (1965), Theil (1973) and Prasada Rao (1972)
log-change index numbers, in a multiplicative form, can be viewed as weighted
geometric means.
-4Wo
Kloek-Theil index
where w. -
i:l [Pill
N
Wo
Wil+Wi2
2
where w.1 :
Theil index
i:l [Pill
where w.1 =
Prasada Rao index =
WilWi2
Wil+Wi2
[Wil+Wi2.
and
Wil
Pi2qi2
- ZPilqi
Pilqil
I
and Wi2 -
£Pi2qi2
Finally there is a class of index numbers formulae where each
number is a function of one or more of the index numbers which are
measures of location of the N price relatives. Famous examples are
Fisher’s index
: WLI2" PI2
LI2+PI2
Drobisch index
Stuvel’s index -
2
LI2-LIq2
2
.2
+
2
2 £Pi2qi2
+
7.Pilqil
where LI2 and PI2 stand for Laspeyres and Paasche price index numbers.
respectively, and L~2 stands for Laspeyres quantity index number (see Banerjee,
1961). Thus statistical approach can be viewed as a fairly useful approach
encompassing most of the index formulae in the literature.
1.2 Welfare approach: This approach draws heavily from the micro-economic
theory concerning a consumer and generally useful in clarifying what a
cost-of-living index number should measure. This approach is also useful
-5-
in making price comparisons after al!owance is made for changes in personal
tastes and quality of goods consumed, which cannot be incorporated into
the index numbers through statistical approach.
This approach starts with the assumption that there is a ’well
behaved’ utility function, or equivalently a proper preference structure,
guiding individual’s decisions regarding consumption. Let U(q) represent
such a utility function, then corresponding to any level U, , one can
evaluate the cost of achieving this utility at a given level of prices
say p, by minimizing the expenditure over all the commodity vectors which
keep the individual on the same level of indiffernce as U, . Let C
represent the cost function then C is given by
n
C(p, U,) : minimum ~ piqi over {q: U(q) : U,}
i=l
= min p’q
{q: U(q) = U,}
Then welfare approach defines the cost-of-living iidex as the ratio of
costs under two different price situations Pl and P2
~12 -
C(P2, U,)
C(Pl, U,)
(1.2.1)
where U.a is a pre-specified level of utility. One may specify U, through
a commodity bundle q, as U, = U(q,).
Now actual construction of index numbers would involve specification
of utility function as well as the utility level at which the index is
defined and this will enable one to determine the costs involved in (1.2.1).
Many index numbers can be derived by identifying the implied functions U
and level of utility U, used in (1.2.1). Popular Laspeyres index can be
seen to be using a fixed coefficient utility function and. U, is U(ql).
¯ --6--
Evidently we assume that the observed pairs (pl,ql) ~ind (p2,q2) are
optimal in situations i and 2. This leads to an index
C(P2,U,)
C(P2,U(ql))
C(P2,U(ql))
I12 - C(PI,U,) - C(PI,U(ql)) = ZplqI,
-
zP2ql
£plqI
Similarly Paasche index can be derived using U, = U(q2) and a fixed
coefficient utility function. This gives
C(P2,U*) _ C(P2’U(q2)) _ ZP2q2
I12 - C(PI,U,)
C(PI,U(q2))
ZPlq2
Both of these follow from the fact that ZplqI and Zp2q2 are optimal in
periods i and 2 respectively.
This can be easily extended to other index numbers, but with a
difference. If we use U, = U(q,) where q, differs from observed ql or q2
then the index defined (1.2.1) requires both the costs to be computed,
which was not the case with Laspeyres or Paasche index numbers~ This
result is true since ql and q2 are optimal in periods i and 2 and the
utility function is of a fixed coefficient type. In other cases, we need
to know the exact ratios involved in the fixed coefficient function to be
able todetermine the two costs involved in (1.2.1). In such cases, those
indexes can be Used as approximations to thetrue index number in (1.2.1).
For example, the Edgeworth-Marshall index can be considered as
=
I12
ZP2(ql+q2)
~
Zpl(ql+q2) --
C(P2, U(ql+q2))
C(Pl, U(ql+q2))
There are many recent studies, like Samuelson and Swamy (1974),
Fishe~ and Shell (1973), Theil (1975), which go into many other aspects
of construction of true-cost-of-living numbers.
-7-
SECTION 2. A new Approach
This approach draws its basic concepts from astudy by Geary (1958)
which were examined and studied by Khamis (1970), Prasada Rao (1970) and
U.N. Study (1973). Geary in his study introduces two concepts, ’exchange
Pate’ and ’average. price’ and uses them to derive an index number formula,
which is now known as Geary-Khamis index number. We develop this new
approach by observing that what Geary-Khamis index number uses in the
definition is only one way of looking at these concepts and many other
possible definitions and interpretations exist and some of them are more
meaningful than the definitions used by Geary and Khamis. We have a brief
discussion of these concepts below..
2.1 Basic Concepts: The concept of ’exchange rate’ seems to be relevant
wheneve~ two situations with differentia! price levels are considered.
Even if the same currency unit is used in the two situations, very common
in inter-temporal comparisons and inter-regional comparisons within a
country, the purchasing power of the currehcy unit is different and hence
they should be treated differently. This is to say that sums of money
expressed in same currency units over time are not strictly additive. In
such cases, the Snly way toadd them is to establish an ’exchange rate’
between these currencies so that all the sums of money can be translated
into sums which are additive. This idea is similar to the idea of exchange
rates for currencies of different nations. So the ’exchange rate’ concept
is a logical off-shoot of the fact that the price levels in different time
periods are different and this results in differentia! purchasing powers
of money. The idea of exchange rate is implicit in al! price comparisons
and index nu~er construction. Obviously the price index I12 would be
the ratio of exchange rates.
The concept of ’average price’ of a commodity over all the
situations crops up as a co~ollary. If the exchange rates are known,
we can translate all the prices into a common currency unit which allows
us to define average price of a commodity, average over all observations.
¯ This would not be meaningful if the exchange rates were not used to make
price observations comparable.
These two concepts are of a fairly general nature and they can
be used for construction of price index numbers for binary as Well as
multilateral comparisons. It seems that these two concepts have a much
wider applicability and conceptually more powerful than what was realized
in the definitions used by Geary and Khamis. In the next few subsections
we consider comparison of general price levels in M situations with (pl,ql),
(p2,q2) .... (pM,qM) representing the price and quantity data. Since we make
distinction between money units corresponding to each pair (pj,qj) We refer
to the money unit as cumrency unit. Let R1, R2,...,RM denote the exchange
rates for different currency units and PI’ P2’ "’’’ PN denote the average
prices of different commodities, stated in a common currency unit, average
over al! the Msets of observations. ¯ We briefly state the Geary-Khamis
system of index numbers and then demonstrate how a few other index numbers
can be derived f~om these concepts. The price index for kth vector with
jth vector as base Ijk is given by Rj/~.
2.2 Gea~y-Khamis system of index numbems: In this subsection we briefly
summamize the Gea~y-Khamis system of index numbers and indicate possible
ways of using this basic structume to derive other index number systems¯
Gear,] (1958) defines the exchange mates and avemage pmices using the following
simultaneous system of linear equations¯ The exchange rate Rj for jth
¯ M, al~l the. average price P. 1of ith commodity,
currency, j = 1,2, ..,
-9-
i : 1,2,...,N, are given by
N
~ Piqi~
R. =
3
i=l
~
N
i!iPijqij
p. = [ RjPijqij/ !
1
j iqij
j=l
’Average price of each commodity is defined using the total value of
ith commodity, expressed in a common currency unit, and the total
quantity of ith commodity, total obtained over all M situations. This
definition makes.use of the implicit price component in a pair of value
and quantity observations. Then the exchange rate for jth currency unit
R]. is defined., by comparing the price vector pj and the average prices
P., i = I,...,N. This is dol,e by comparing the expenditure sufficient
l
tO buy the quantity vector qj at these two price vectors. Thus the
equati0n, system (2.2.1) seems to be a system derived from intuitive ’
understanding of the concepts of exchange rates and average prices.
Usefulness of equation system (2.2.1) for price and quantity
comparisons depends upon whether meaningful solution Of R.’s and P.’s
emerges from this for any set of , plausible, price and quantity vectors.
This property has been established in Prasada Rao (1970) where
necessary and sufficient conditions for the existence of unique (up to
scalar multiplication) positive solutions for R.’s] and P.’s
were derived.
1
Solution f~r R.’s for the case M = 2 is given by
]
N
[ Pii qi2qil
i=l
qi2+qil
RI = i and R2 =
N
~ Pi2 qi2qil
i=l
qi2+qil
(2.2.1)
-i0-
and the index I12, given by RI/R2 would be
I12 =
N
[ Pi2
i=l
N
[ Pil
i=l
qi2qil
qi2+qil
qi2qil
qi2+qil
this index was already listed in Section i.
Now we derive some of the other index numbers known in the
literature. Before we do this let us look at equation system (2.2.1).
It is eisy to see that the exchange rate Rj is defined as a weighted-
N
average of (Pi/Pij), i = I,...,N, with weights Pijqij)i=l
~ pi~qij
and
~
P. is defined as a weighted average of (R. P..), j = I, ..,M, with weights
~
] ~]
¯
M
~
qi~/ ~ qi~.] Now it is evident that (2.2.1) is not a unique definition for
J j:l
system of index numbers since we can obtain different systems by using
different averaging procedures and weighing schedules.
Laspeyres, Paasche and Fisher Index Numbers:
Consider the following set of equations ~efining R.’s
and P’s
i
]
R. - £PiqiJ
] EPijqij
for j = I,...,M
(2.3.1)
. =
P~
1
-M ZRp
j ij
for i = I,...,N
In the system (2.3.1), the exchange rates are defined in the
same manner as in (2.2.1), but the average price of each commodity is
defined as simple average of the ith commodity prices in M situations
transformed into a common currency. Let us look at the solution for
-iiR.’s from (2.3.1) with M = 2 since we are interested in Laspeyres, Paasche
3
and~ Fisher indices which are defined for binary comparisons. Substituting
the values of P.’s in R.’s, we have fork= i, 2..
3
~
i=l
g
qik
j
~ Pikqik
i=l
N
N 2
1
M
N
+
= M ~. Pikqik
~ Pikqik
i=l
i Thus we have RI(I- ~)
i=l
i R2i!iPi2qik
-M N
~ Pikqik
i=l
R2 ~1ZPilqil
ZPi2qil - 0
_ R1 1 ZPilqi2 + R2(I~) = 0
¯
M ZPi2qi2
or, since M = 2,
ZPi2qil
ZPilqil
ZPilqi2
ZPi2qi2
Obviously this system has only a trivial solution. But to obtain meaningful
price comparisons we require positive solutions. This can be done by
ignoring one of the equations in (~.3.2) and solve for R1 and R2.
Let us look at this alternative more closely Obviously ignoring
one ofthe equations would imply ignoring the quantities in one of the
-14-
equations are dropped. Now if we are interested in an index Ijk then
we ignore the current quantity vector % by dropping the kth equation
from the system and define Ijk as ~/Rj where ~ and R.] are obtained
as solutions from the system without kth equation. Similarly Paasche
index can be defined. Analagous to the case M = 2, we can derive a new
set of exchange rates using all the diffirent exchange rate vectors by
taking the geometric mean. These exchange rates, give generalized Fisher’s
index numbers. Interpreting and deriving these most often used index
numbers using the exchange rate and average prices can lead to a straight
forward and meaningful way of generalizing these index number~ -for multilateral comparisons without using the idea of chain index numbers.
2.4
ECLA or Fixsd Weight Index Numbers:
Fixed weight index numbers are the simplest index numbers for
binary or multilateral comparison. Let qa be an arbitrary vector of
quantities, may or may not depend on the observed quantity vectors qj,
j : i,..., M. There are many interesting special cases of these fixed
qil+qi2
for each i,
weight index numbers. For the case M = 2, qia
2
leads to the Edgeworth-Marsha.~index
£Pi2(qil+qi2) .
112 = EPil(qil+qi2)
Similarly, qia = /qi2qil give rise to Drobisch index
EPi2 q/~i2qil .
I12 = £Pil ~
The index number used to compute prices indices for Latin American
= ~i and the indices
M
i J!lqiJ
=
¯ countries, Braithwaite (1968) uses qia ~
are given by _
-15E Pikqia
ljk- EPijqi
a
£ Pikqi
- ~Pij~i
(2.4.i)
These indices can also be derived using the exchange rate framework.
The following system of equations leads to the ECLA indices
N
i=i
~ Pij~i
i=l
Pi =
M
[ RjPijqij
j=l
M
.I qij
]=i
(2.4.2)
This system obviously gives index number ljk identica! to the index in
(2.4.1). However even if we replace Pi by any other definition still
the ihdices that result in would be identical to the one in (2.4.2) and
(2.4.1). As a corollory, Edgeworth-MarshaL and Drobisch indices can
be derived f~om (2.4.2) by using appropriate ~i"
2.5 Kloek-Theil Index Numbers:
~
So far we have been looking at only index numbers which are in
an additive fo~m. Now we look into some index numbers which involve
products and geometric means. Kloek-Theil (1965) used the following
index number for international pride comparison with M = 2. This binary
index I12 is defined as
N ~pi~] Vil+Vi2/2
= H
112
i=l ~Pil
where v..’s
=
~] are the expenditure shames and v..~]
Pijqij
N
i!’iPijqij
-16for i : i,..., N and j : 1,2
This is obviously a weighted geometric
mean of the price relatives wheme the weight foe each item is the average
of the expenditure shames.
We will derive this system of index numbems using exchange mate
and average price concepts with M = 2.~ Consider
N [ Pi ] 1]
R. : ~
-3 i=l
Pij
j = I,..., M
(2.5.1)
i
M
P : ~ [R.p..] ~
i j:l ]l]
i = i,..., N
It is easy to see that (2.5.1) is infact a multiplicative version of
(2.3.1), which gave Laspeymes and Paasche index numbers. This gives an
implicit relationship between Laspeyres, Paasche, Fisher and Theil-Kloek
index numbers,
Transforming (2.5.1) into logarithms we have for M = 2,
N
log R.] : i:l~ (log Pi - log
Pij4) v..x]
j~: 1,2
1 (log R. + log
)
log Pi : ~
Pij
]
i : 1,2, .... , N.
This yields a linear non-homogeneous equation system in log R.’s. By
]
eliminating P.’s we have
l
2 N
,
log Pij) Vil].
(2.s.2)
½ ~. ~ (log Pij
½ Llog R2
j:l i:1
log P2j)
vi2]
In this section we present a system which is consistent and yields index
numbers which are consistent for multilateral comparisons. This is
achieved by a slight.modification of the system (2.5.1). We propose a new
system which makes use of a weighted geometric average of RjPij , ’s weights
depending on the value ratios. Consider the system
j = I,...,M
and
M
v..
p. --
PiJqiJ
N
v.. :
where
,.
vii____
- and v.. :
j :l
Definition of average prices used in (2.6..1) is the multiplicative
vers ion o f
where each R.
is weighted by the relative importance "
of the
] Pij
’
expenditure share.
If we transform system (2.6.1) into a linear form using logarithm
and eliminating Pi’s we get for K = 1,2,..., M
M N
*
M N * ":~
¯ :~
]
+ I I [log Pij - log Pik VikVi~
log Rk : [ [ R.
~=li=l ] vi~ vik ~=I i=l
where
(2.6.2)
*
Re:i !iPikqik
for each k.
-19Thisis a set of M linear non-homogeneous equations in as many unknowns.
Prasada Rao (1972) proves that this system is consistent, i.e., there
exists a solution for ~’s and hence ~’s for this equation system and
that the solution is unique up to scalar multiplication, given that
price and quantity data satisfy same regularity conditions as required for
Geary-Khamis method.
Thus we have a viable system of index numbers from yet another
set of definitions of exchange rates and average prices. This system
seems to have two natural advantages over other systems cbnsidered so
far including the Geary-Khamis system. Since this system uses weighted
geometric means this is more robust towards the presence of illbehaved
quantity and price vectors. Such vectors can indeed produce unacceptable
results when we use Geary-Khamis and other consistentmethods. Further
average prices in (2.6.1) are defined with weights depending on the
expenditure shares rather than the quantity shares. So the average
prices (2.6.1).fluctuate less violently than the systems which depend
on quantity weights.
Let us !ook at the resulting index numbers in the case M = 2.
solution for RI and R2 is given by
The
v"
Vil i2"i=
1 il i2
RI = i
and R2 : i=l
H
Pi2J
N
and the price index is
VilVi2 i=lvilvi2
i:l [Pill
This index can be seen to be a weighted geometric mean of the price
relatives with harmonic mean ofthe expenditure shares Vil and vi2 as
weights where as the Kloek-Theil index uses arithmetic mean of expenditure
shares as weights.
REFERENCES
Banerjee, K.S. (1961): ’A unified statistical approach to the index~
number problem’, Econometrica, Voi.29.
Braithwaite, S.N. (1968): ’Real Income levels in Latin America’, Review
of Income and Wealth, Series i;~, No. 2.
Fisher, I. (1922): ’The making of Index Numbers’, Houghton Miffin, Boston,
Mass.
Fisher, F.M. and Shell, K. (1973): ’Economic Theory of Price Indexes’
Academic Press.
Geary, R.C. (1958): ’A note on the comparison of exchange rates and
purchasing powers between countries’, J.R.S.S.,
Vol. 121.
Khamis , S.H. (1972): ’A new system of Index Numbers for National and "
International purposes’ J R.S.S. Series A
Kloek, T. and Theil, H. (1965): ’International comparisons of prices and
quantities consumed’ Econometrica, Vol 33
Prasa~a Rao, D.S.~ (1970) ’Existence and uniqueness of anew class of index
numbers’, Sankhya~ Series B, Vol. 33.
Prasada Ra0, D.S. (1972): Contributions to Methodology of construction of
Consistent index numbers. Unpublished Ph.D. Dissertation, Indian
Statisticai Institute.
Prasada Rao, D.S. (1976): ’Existence and Uniqueness of a system of Consistent
Index Numbers’, Economic Review, (Japan), Vol. 7.
Samuelson, P. and Swamy, S. (1974): ’Invariant Economic Index Numbers and
canonical duality. Survey and Synthesis’ American Economic Review.
Theil, H. (1973): ’A new index number formula’, Review of Econom~s and
Statistics.
Theil, H. (1975): ’Theory and Measurement of consumer demand Vol i’
North-Holland Publishing company.
United Nations (1973): International Price comparisons project methods for
International Product and Purchasing Power comparisons. Statistical
Office of U.N.